Totally flat foliations and peakless functions
A continuous function is peakless if it is monotone on each geodesic except possibly on an interval where it attains a minimum. Suppose in addition the function goes to infinity in any direction in which it is increasing. Let \(\{f_ i\}\) be a sequence of functions on a noncompact space satisfying the above conditions, and such that for some constants \(\{a_ i\}\), the sets where \(f_ i\leq a_ i\) are pairwise disjoint and the complement of their union is bounded. Then the space has exactly two ends, defined by two suitably chosen functions of the sequence, and the boundaries of the sets where \(f_ i\leq a_ i\) define a totally geodesic foliation of part of the space. The result is obtained in the context of G-spaces, and assumes no differentiability.
scientific article; zbMATH DE number 3883163
scientific article